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Theorem uneq12i 3442
Description: Equality inference for union of two classes. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
Hypotheses
Ref Expression
uneq1i.1  |-  A  =  B
uneq12i.2  |-  C  =  D
Assertion
Ref Expression
uneq12i  |-  ( A  u.  C )  =  ( B  u.  D
)

Proof of Theorem uneq12i
StepHypRef Expression
1 uneq1i.1 . 2  |-  A  =  B
2 uneq12i.2 . 2  |-  C  =  D
3 uneq12 3439 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  u.  C
)  =  ( B  u.  D ) )
41, 2, 3mp2an 654 1  |-  ( A  u.  C )  =  ( B  u.  D
)
Colors of variables: wff set class
Syntax hints:    = wceq 1649    u. cun 3261
This theorem is referenced by:  indir  3532  difundir  3537  difindi  3538  symdif1  3549  unrab  3555  rabun2  3563  dfif6  3685  dfif3  3692  dfif5  3694  unopab  4225  xpundi  4870  xpundir  4871  xpun  4875  dmun  5016  resundi  5100  resundir  5101  cnvun  5217  rnun  5220  imaundi  5224  imaundir  5225  dmtpop  5286  coundi  5311  coundir  5312  unidmrn  5339  dfdm2  5341  mptun  5515  resasplit  5553  fresaun  5554  fresaunres2  5555  fpr  5853  fvsnun2  5868  sbthlem5  7157  1sdom  7247  cdaassen  7995  fzo0to42pr  11113  hashgval  11548  hashinf  11550  vdwlem6  13281  setsres  13422  xpsc  13709  lefld  14598  opsrtoslem1  16471  volun  19306  iblcnlem1  19546  usgraexvlem  21280  ex-dif  21579  ex-in  21581  ex-pw  21585  ex-xp  21592  ex-cnv  21593  ex-rn  21596  partfun  23928  indval2  24208  sigaclfu2  24300  subfacp1lem1  24644  subfacp1lem5  24649  predun  25214  symdif0  25392  symdifid  25394  fixun  25473  bpoly3  25818  onint1  25913  itg2addnclem2  25958  itgaddnclem2  25964  iblabsnclem  25968  refssfne  26065
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-v 2901  df-un 3268
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